Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 43-46: The given equation has the form \(y=f(x)\). (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Solve for \(x\) in terms of \(y\). $$ y=4 \cos ^{-1} \frac{2}{3} x $$

Step-by-Step Solution

Answer

Domain: \([-\frac{3}{2}, \frac{3}{2}]\); Range: \([0, 4\pi]\); \(x = \frac{3}{2} \cos\left( \frac{y}{4} \right)\).

Step 1: Identify the function form and constraints

The given function is \( y = 4 \cos^{-1}\left( \frac{2}{3} x \right) \). The function involves the inverse cosine, \( \cos^{-1} \), which requires that the input, \( \frac{2}{3}x \), must be within the domain of \( [-1, 1] \).

Step 2: Find the domain of the function

The domain of \( \cos^{-1}(u) \) is \( -1 \leq u \leq 1 \). For \( u = \frac{2}{3}x \), the domain constraint is: \[-1 \leq \frac{2}{3}x \leq 1.\]Solving these inequalities: 1. \( \frac{2}{3}x \geq -1 \) implies \( x \geq -\frac{3}{2} \). 2. \( \frac{2}{3}x \leq 1 \) implies \( x \leq \frac{3}{2} \). Hence, the domain of \( f(x) \) is \( [-\frac{3}{2}, \frac{3}{2}] \).

Step 3: Find the range of the function

The range of \( \cos^{-1}(u) \) is \([0, \pi]\), so for \( y = 4 \cos^{-1} \frac{2}{3} x \), the range modifies to:\[ 0 \cdot 4 \leq y \leq 4\pi.\]Thus, the range of \( f(x) \) is \([0, 4 \pi]\).

Step 4: Solve for \( x \) in terms of \( y \)

Starting with the given equation:\[y = 4 \cos^{-1} \left( \frac{2}{3}x \right).\]Divide both sides by 4:\[\cos^{-1} \left( \frac{2}{3}x \right) = \frac{y}{4}.\]Taking the cosine of both sides results in:\[\frac{2}{3}x = \cos\left( \frac{y}{4} \right).\]Multiply by \( \frac{3}{2} \):\[x = \frac{3}{2} \cos\left( \frac{y}{4} \right).\]This expression gives \( x \) in terms of \( y \).

Key Concepts

Understanding Domain and Range

When dealing with inverse trigonometric functions, it's crucial to identify the domain and range. Here, the function involves the inverse cosine, denoted as \( \cos^{-1} \), which has specific restrictions.

• The **domain** of \( \cos^{-1}(u) \) is \([-1, 1]\). This implies that the input, in our function, represented by \( \frac{2}{3}x \), must be constrained within this interval.
• To find the domain of \( f(x) \), we solve the inequality for \( \frac{2}{3}x \):
  • \( -1 \leq \frac{2}{3}x \leq 1 \).
  • After solving this, we get \( x \) must be between \(-\frac{3}{2}\) and \(\frac{3}{2}\).
  • Thus, the domain is \([-\frac{3}{2}, \frac{3}{2}]\).
• The **range** of \( \cos^{-1}(u) \) is \([0, \pi]\). Since our function involves a multiplication by 4, the range becomes \([0, 4\pi]\).

Recognizing these constraints helps in determining valid inputs and possible outputs for the function.

Solving Equations Involving Inverse Trigonometric Functions

To solve equations involving inverse trigonometric functions like \( \cos^{-1} \), follow these steps. Starting with the equation \( y = 4 \cos^{-1} \left( \frac{2}{3}x \right) \), the goal is to find \( x \) in terms of \( y \).

• First, isolate the inverse cosine function: divide both sides by 4. This gives \( \cos^{-1} \left( \frac{2}{3}x \right) = \frac{y}{4} \).
• Next, to eliminate the inverse cosine, apply the cosine to both sides of the equation:
  • This results in \( \frac{2}{3}x = \cos\left( \frac{y}{4} \right) \).
• The final step is solving for \( x \):
  • Multiply by \( \frac{3}{2} \) to get \( x = \frac{3}{2} \cos\left( \frac{y}{4} \right) \).

These steps demonstrate how to manipulate inverse trigonometric equations to express one variable in terms of another.

Trigonometric Identities and Their Uses

Trigonometric identities can simplify equations and integrate inverse trigonometric functions. While the specific problem doesn't directly apply common identities like the Pythagorean identity or angle sum formulas, understanding them can provide valuable insight.

• **Basic Identity**: Remember that \( \cos^{2}(x) + \sin^{2}(x) = 1 \). This identity is fundamental when working with any trigonometric problem.
• **Inverse Relationships**: Since we're using \( \cos^{-1} \), recall that \( \cos(\cos^{-1}(u)) = u \) for \( u \) in \([-1, 1]\).
• **Applications**: While simpler identities aren't used in the function \( y = 4 \cos^{-1} \left( \frac{2}{3}x \right) \), knowing these relationships helps verify solutions and make predictions about the behavior of trigonometric expressions.

Grasping these identities builds a strong foundation for solving more complex inverse trigonometric functions in the future.